检查给定号码是否为欧几里得号码

给定一个正整数n，任务是检查它是否为Euclid Number。如果给定的号码是欧几里德号码，则打印“ YES”，否则打印“ NO”。

欧几里德数：在数学中，欧几里德数是以下形式的整数： $E_{n} = p_{n}\# + 1$
在哪里$p_{n}\#$是前n个质数的乘积。

前几个Euclid号码是-

3, 7, 31, 211, 2311, 30031, 510511, 9699691, ……….

例子：

Input: N = 31
Output: YES
31 can be expressed in the form of 
pn# + 1 as p3# + 1
(First 3 prime numbers are 2, 3, 5 and their product is 30 )

Input: N = 43
Output: NO
43 cannot be expressed in the form of pn# + 1

天真的方法：

1. 使用Eratosthenes筛网生成该范围内的所有素数。
2. 然后从第一个质数(即2)开始乘以下一个质数，并继续检查乘积+ 1 = n 。
3. 如果乘积+1 = n，则n是欧几里德数。否则不行。

下面是上述方法的实现：

C++

// CPP program to check Euclid Number
  
#include 
using namespace std;
  
#define MAX 10000
  
vector arr;
  
// Function to generate prime numbers
void SieveOfEratosthenes()
{
    // Create a boolean array "prime[0..n]" and initialize
    // all entries it as true. A value in prime[i] will
    // finally be false if i is Not a prime, else true.
    bool prime[MAX];
    memset(prime, true, sizeof(prime));
  
    for (int p = 2; p * p < MAX; p++) {
        // If prime[p] is not changed, then it is a prime
  
        if (prime[p] == true) {
  
            // Update all multiples of p
            for (int i = p * 2; i < MAX; i += p)
                prime[i] = false;
        }
    }
  
    // store all prime numbers
    // to vector 'arr'
    for (int p = 2; p < MAX; p++)
        if (prime[p])
            arr.push_back(p);
}
  
// Function to check the number for Euclid Number
bool isEuclid(long n)
{
  
    long long product = 1;
    int i = 0;
  
    while (product < n) {
  
        // Multiply next prime number
        // and check if product + 1 = n
        // holds or not
        product = product * arr[i];
  
        if (product + 1 == n)
            return true;
  
        i++;
    }
  
    return false;
}
  
// Driver code
int main()
{
  
    // Get the prime numbers
    SieveOfEratosthenes();
  
    // Get n
    long n = 31;
  
    // Check if n is Euclid Number
    if (isEuclid(n))
        cout << "YES\n";
    else
        cout << "NO\n";
  
    // Get n
    n = 42;
  
    // Check if n is Euclid Number
    if (isEuclid(n))
        cout << "YES\n";
    else
        cout << "NO\n";
  
    return 0;
}

Java

// Java program to check Euclid Number
  
import java.util.*;
  
class GFG {
  
    static final int MAX = 10000;
    static Vector arr = new Vector();
  
    // Function to get the prime numbers
    static void SieveOfEratosthenes()
    {
        // Create a boolean array "prime[0..n]" and initialize
        // all entries it as true. A value in prime[i] will
        // finally be false if i is Not a prime, else true.
        boolean[] prime = new boolean[MAX];
  
        for (int i = 0; i < MAX; i++)
            prime[i] = true;
  
        for (int p = 2; p * p < MAX; p++) {
  
            // If prime[p] is not changed, then it is a prime
            if (prime[p] == true) {
  
                // Update all multiples of p
                for (int i = p * 2; i < MAX; i += p)
                    prime[i] = false;
            }
        }
  
        // store all prime numbers
        // to vector 'arr'
        for (int p = 2; p < MAX; p++)
            if (prime[p])
                arr.add(p);
    }
  
    // Function to check the number for Euclid Number
    static boolean isEuclid(long n)
    {
  
        long product = 1;
        int i = 0;
        while (product < n) {
  
            // Multiply next prime number
            // and check if product + 1 = n
            // holds or not
            product = product * arr.get(i);
  
            if (product + 1 == n)
                return true;
  
            i++;
        }
  
        return false;
    }
    public static void main(String[] args)
    {
  
        // Get the prime numbers
        SieveOfEratosthenes();
  
        // Get n
        long n = 31;
  
        // Check if n is Euclid Number
        if (isEuclid(n))
            System.out.println("YES");
        else
            System.out.println("NO");
  
        // Get n
        n = 42;
  
        // Check if n is Euclid Number
        if (isEuclid(n))
            System.out.println("YES");
        else
            System.out.println("NO");
    }
}

Python 3

# Python 3 program to check 
# Euclid Number
MAX = 10000
  
arr = []
  
# Function to generate prime numbers
def SieveOfEratosthenes():
  
    # Create a boolean array "prime[0..n]" 
    # and initialize all entries it as 
    # true. A value in prime[i] will
    # finally be false if i is Not a
    # prime, else true.
    prime = [True] * MAX
  
    p = 2
    while p * p < MAX :
          
        # If prime[p] is not changed, 
        # then it is a prime
        if (prime[p] == True):
  
            # Update all multiples of p
            for i in range(p * 2, MAX, p):
                prime[i] = False
                  
        p += 1
  
    # store all prime numbers
    # to vector 'arr'
    for p in range(2, MAX):
        if (prime[p]):
            arr.append(p)
  
# Function to check the number
# for Euclid Number
def isEuclid(n):
  
    product = 1
    i = 0
  
    while (product < n) :
  
        # Multiply next prime number
        # and check if product + 1 = n
        # holds or not
        product = product * arr[i]
  
        if (product + 1 == n):
            return True
  
        i += 1
  
    return False
  
# Driver code
if __name__ == "__main__":
  
    # Get the prime numbers
    SieveOfEratosthenes()
  
    # Get n
    n = 31
  
    # Check if n is Euclid Number
    if (isEuclid(n)):
        print("YES")
    else:
        print("NO")
  
    # Get n
    n = 42
  
    # Check if n is Euclid Number
    if (isEuclid(n)):
        print("YES")
    else:
        print("NO")
  
# This code is contributed 
# by ChitraNayal

C#

// C# program to check Euclid Number
using System;
using System.Collections.Generic;
  
class GFG 
{
  
    static readonly int MAX = 10000;
    static List arr = new List();
  
    // Function to get the prime numbers
    static void SieveOfEratosthenes()
    {
        // Create a boolean array 
        // "prime[0..n]" and initialize
        // all entries it as true. 
        // A value in prime[i] will
        // finally be false if i is
        // Not a prime, else true.
        bool[] prime = new bool[MAX];
  
        for (int i = 0; i < MAX; i++)
            prime[i] = true;
  
        for (int p = 2; p * p < MAX; p++)
        {
  
            // If prime[p] is not changed,
            // then it is a prime
            if (prime[p] == true) 
            {
  
                // Update all multiples of p
                for (int i = p * 2; i < MAX; i += p)
                    prime[i] = false;
            }
        }
  
        // store all prime numbers
        // to vector 'arr'
        for (int p = 2; p < MAX; p++)
            if (prime[p])
                arr.Add(p);
    }
  
    // Function to check the number for Euclid Number
    static bool isEuclid(long n)
    {
  
        long product = 1;
        int i = 0;
        while (product < n) 
        {
  
            // Multiply next prime number
            // and check if product + 1 = n
            // holds or not
            product = product * arr[i];
  
            if (product + 1 == n)
                return true;
  
            i++;
        }
  
        return false;
    }
      
    // Driver code
    public static void Main(String[] args)
    {
  
        // Get the prime numbers
        SieveOfEratosthenes();
  
        // Get n
        long n = 31;
  
        // Check if n is Euclid Number
        if (isEuclid(n))
            Console.WriteLine("YES");
        else
            Console.WriteLine("NO");
  
        // Get n
        n = 42;
  
        // Check if n is Euclid Number
        if (isEuclid(n))
            Console.WriteLine("YES");
        else
            Console.WriteLine("NO");
    }
}
  
// This code has been contributed by 29AjayKumar

输出：

YES
NO

注意：以上方法对于每个查询(每N个)将采用O(P _n ＃) ，即否。要乘以素数的整数，以检查n是否为欧几里德数。

高效方法：

1. 使用Eratosthenes筛网生成该范围内的所有素数。
2. 计算质数的前缀乘积，最大范围应避免使用哈希表重新计算乘积。
3. 如果乘积+1 = n，则n是欧几里德数。否则不行。

下面是上述方法的实现：

C++

// CPP program to check Euclid Number
  
#include 
using namespace std;
  
#define MAX 10000
  
unordered_set s;
  
// Function to generate the Prime numbers
// and store their products
void SieveOfEratosthenes()
{
    // Create a boolean array "prime[0..n]" and initialize
    // all entries it as true. A value in prime[i] will
    // finally be false if i is Not a prime, else true.
    bool prime[MAX];
    memset(prime, true, sizeof(prime));
  
    for (int p = 2; p * p < MAX; p++) {
        // If prime[p] is not changed, then it is a prime
  
        if (prime[p] == true) {
  
            // Update all multiples of p
            for (int i = p * 2; i < MAX; i += p)
                prime[i] = false;
        }
    }
  
    // store prefix product of prime numbers
    // to unordered_set 's'
    long long int product = 1;
  
    for (int p = 2; p < MAX; p++) {
  
        if (prime[p]) {
  
            // update product by multipying
            // next prime
            product = product * p;
  
            // insert 'produc+1' to set
            s.insert(product + 1);
        }
    }
}
  
// Function to check the number for Euclid Number
bool isEuclid(long n)
{
  
    // Check if number exist in
    // unordered set or not
    // If exist, return true
    if (s.find(n) != s.end())
        return true;
    else
        return false;
}
  
// Driver code
int main()
{
  
    // Get the prime numbers
    SieveOfEratosthenes();
  
    // Get n
    long n = 31;
  
    // Check if n is Euclid Number
    if (isEuclid(n))
        cout << "YES\n";
    else
        cout << "NO\n";
  
    // Get n
    n = 42;
  
    // Check if n is Euclid Number
    if (isEuclid(n))
        cout << "YES\n";
    else
        cout << "NO\n";
  
    return 0;
}

Java

// Java program to check Euclid Number
import java.util.*;
  
class GFG 
{
static int MAX = 10000;
  
static HashSet s = new HashSet();
  
// Function to generate the Prime numbers
// and store their products
static void SieveOfEratosthenes()
{
    // Create a boolean array "prime[0..n]" and 
    // initialize all entries it as true. 
    // A value in prime[i] will finally be false 
    // if i is Not a prime, else true.
    boolean []prime = new boolean[MAX];
    Arrays.fill(prime, true);
    prime[0] = false;
    prime[1] = false;
    for (int p = 2; p * p < MAX; p++)
    {
        // If prime[p] is not changed, 
        // then it is a prime
        if (prime[p] == true) 
        {
  
            // Update all multiples of p
            for (int i = p * 2; i < MAX; i += p)
                prime[i] = false;
        }
    }
  
    // store prefix product of prime numbers
    // to unordered_set 's'
    int product = 1;
  
    for (int p = 2; p < MAX; p++) 
    {
        if (prime[p])
        {
  
            // update product by multipying
            // next prime
            product = product * p;
  
            // insert 'produc+1' to set
            s.add(product + 1);
        }
    }
}
  
// Function to check the number for Euclid Number
static boolean isEuclid(int n)
{
  
    // Check if number exist in
    // unordered set or not
    // If exist, return true
    if (s.contains(n))
        return true;
    else
        return false;
}
  
// Driver code
public static void main(String[] args)
{
    // Get the prime numbers
    SieveOfEratosthenes();
  
    // Get n
    int n = 31;
  
    // Check if n is Euclid Number
    if (isEuclid(n))
        System.out.println("Yes");
    else
        System.out.println("No");
  
    // Get n
    n = 42;
  
    // Check if n is Euclid Number
    if (isEuclid(n))
        System.out.println("Yes");
    else
        System.out.println("No");
}
} 
  
// This code is contributed by PrinciRaj1992

Python3

# Python3 program to check Euclid Number 
MAX = 10000
  
s = set() 
  
# Function to generate the Prime numbers 
# and store their products 
def SieveOfEratosthenes(): 
  
    # Create a boolean array "prime[0..n]" 
    # and initialize all entries it as true. 
    # A value in prime[i] will finally be
    # false if i is Not a prime, else true. 
    prime = [True] * (MAX) 
    prime[0], prime[1] = False, False
      
    for p in range(2, 100):
          
        # If prime[p] is not changed, 
        # then it is a prime 
        if prime[p] == True: 
  
            # Update all multiples of p 
            for i in range(p * 2, MAX, p): 
                prime[i] = False
  
    # store prefix product of prime numbers 
    # to unordered_set 's' 
    product = 1
  
    for p in range(2, MAX): 
  
        if prime[p] == True:
  
            # update product by multipying 
            # next prime 
            product = product * p 
  
            # insert 'produc+1' to set 
            s.add(product + 1) 
  
# Function to check the number 
# for Euclid Number 
def isEuclid(n):
  
    # Check if number exist in 
    # unordered set or not 
    # If exist, return true 
    if n in s: 
        return True
    else:
        return False
  
# Driver code 
if __name__ == "__main__":
  
    # Get the prime numbers 
    SieveOfEratosthenes() 
  
    # Get n 
    n = 31
  
    # Check if n is Euclid Number 
    if isEuclid(n) == True: 
        print("YES") 
    else:
        print("NO") 
  
    # Get n 
    n = 42
  
    # Check if n is Euclid Number 
    if isEuclid(n) == True: 
        print("YES") 
    else:
        print("NO") 
  
# This code is contributed by Rituraj Jain

C＃

// C# implementation of the approach
using System;
using System.Collections.Generic;
      
class GFG 
{
static int MAX = 10000;
static HashSet s = new HashSet();
  
// Function to generate the Prime numbers
// and store their products
static void SieveOfEratosthenes()
{
    // Create a boolean array "prime[0..n]" and 
    // initialize all entries it as true. 
    // A value in prime[i] will finally be false 
    // if i is Not a prime, else true.
    Boolean []prime = new Boolean[MAX];
    for (int p = 0; p < MAX; p++)
        prime[p] = true;
    prime[0] = false;
    prime[1] = false;
    for (int p = 2; p * p < MAX; p++)
    {
        // If prime[p] is not changed, 
        // then it is a prime
        if (prime[p] == true) 
        {
  
            // Update all multiples of p
            for (int i = p * 2; i < MAX; i += p)
                prime[i] = false;
        }
    }
  
    // store prefix product of prime numbers
    // to unordered_set 's'
    int product = 1;
  
    for (int p = 2; p < MAX; p++) 
    {
        if (prime[p])
        {
  
            // update product by multipying
            // next prime
            product = product * p;
  
            // insert 'produc+1' to set
            s.Add(product + 1);
        }
    }
}
  
// Function to check the number 
// for Euclid Number
static Boolean isEuclid(int n)
{
  
    // Check if number exist in
    // unordered set or not
    // If exist, return true
    if (s.Contains(n))
        return true;
    else
        return false;
}
  
// Driver code
public static void Main(String[] args)
{
    // Get the prime numbers
    SieveOfEratosthenes();
  
    // Get n
    int n = 31;
  
    // Check if n is Euclid Number
    if (isEuclid(n))
        Console.WriteLine("Yes");
    else
        Console.WriteLine("No");
  
    // Get n
    n = 42;
  
    // Check if n is Euclid Number
    if (isEuclid(n))
        Console.WriteLine("Yes");
    else
        Console.WriteLine("No");
}
}
  
// This code is contributed by Princi Singh

输出：

YES
NO

注意：以上方法将花费O(1)时间来回答查询。